No CrossRef data available.
Article contents
Further Duals of a verbal Subgroup
Published online by Cambridge University Press: 18 May 2009
Extract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
In a previous paper [3] we gave two methods for constructing subgroups which in certain senses may be considered to be dual to a verbal subgroup Vf(G) of an arbitrary group G. Associated with a word h (u, v) in the two symbols u and v, we have (i) the first dual subgroup 
which is defined as the minimal subgroup of G containing all elements ξ of G for which
for all values of x1, x2, …, in xn in G, and (ii), the second dual subgroup 
which is defined as the minimal subgroup of G containing all elements z of G for which
for all values of x1, x2, …, xn in G. Below we introduce slight variations to these definitions, which give rise to the concepts of the third and the fourth dual subgroups respectively. For certain values of h(u, v) we obtain concepts which also arise from and , namely, the marginal subgroup, the invariable subgroup and the centralizer of a verbal subgroup. We also obtain the new concepts of elemental subgroups and commutal subgroups and briefly sketch some of their properties. Finally we conclude by showing that MacLane's dual for the centralizer of a verbal subgroup is a closely related verbal subgroup.
- Type
- Research Article
- Information
- Copyright
- Copyright © Glasgow Mathematical Journal Trust 1959
References
1.MacLane, S., Groups, categories and duality, Proc. Nat. Acad. Sci., 34 (1948), 263–267.Google Scholar
3.Moran, S., Duals of a verbal subgroup, J. London Math. Soc., 33 (1958), 220–236; Corrigenda, 34(1959), 250.CrossRefGoogle Scholar
4.Moran, S., Associative operations on groups III, Proc. London Math. Soc. (3) 9 (1959), 287–317.CrossRefGoogle Scholar
5.Witt, E., Treue Darstellung Liesche Ringe, J. reine angew. Math. 177 (1937), 152–160.Google Scholar
You have
Access